[637 
638] 
25 
IY OF ELLIPTIC 
p. 29.] 
, writing Q—x and 
9), 
638. 
ON A (¿-FORMULA LEADING TO AN EXPRESSION FOR E,. 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 63—66.] 
It is to be shown that we have identically 
(1 + + 2+ 2j* + ...)< - 16 (j^+ n^ + riy + -) 
1 - 9q - 25q 3 + 49g 6 + 81 g lu - ... 
1 - q — q s + q 6 + q w — ... 
or, what is the same thing, 
(1 _ 2s + 2s* - 2j* + .. .)* - 16 + ...) 
_ 1 + 9</ + %5q 3 + 49q 6 4- SI«? 10 + ... 
i + q + q 3 + q 6 + q 10 + ■■■ 
(A); 
(B), 
where the form (A) is that intended to be made use of, but the form (B) is rather 
more convenient for the demonstration. 
We have 
(1-2S + 2S 4 —2s* + ...)<=l + 8{ i ^ + I ^-^l + ...} 
(Jacobi, Fund. Nova, p. 188, Ges. Werke, t. i., p. 239), taking the formula as there 
written down, and changing q into — q. 
Also, if for a moment 
and 
X = 1 + q + q 3 + f + q w + & c., 
r, dX 
X' 
dq 
C. X. 
4